Bait and Ditch: Consumer Naïveté andSalesforce Incentives∗

Fabian Herweg† Antonio Rosato‡

December 19, 2018

Abstract

We analyze a model of price competition between a transparent retailer and a deceptive

one in a market where a fraction of consumers is naïve. The transparent retailer is an

independent shop managed by its owner. The deceptive retailer belongs to a chain and is

operated by a manager. The two retailers sell an identical base product, but the deceptive

one also offers an add-on. Rational consumers never consider buying the add-on while naïve

ones can be “talked”into buying it. By offering its store manager a contract that pushes him

to never sell the base good without the add-on, the chain can induce an equilibrium in which

both retailers obtain more-than-competitive profits. The equilibrium features price dispersion

and market segmentation, with the deceptive retailer targeting only naïve consumers whereas

the transparent retailer serves only rational ones. Consumer welfare is not monotone in the

fraction of naïve consumers. Hence, policy interventions designed to de-bias naïve consumers

might actually backfire.

JEL classification: D03, D18, D21, L13, M52.

Keywords: Add-On Pricing; Bait and Switch; Consumer Naïveté; Incentive Contracts.

∗We thank Juan Carlos Carbajal, Paul Heidhues, Johannes Johnen, Botond Koszegi, Shiko Maruyama, TakeshiMurooka, Larry Samuelson, Emil Temnyalov, Tom Wilkening, and seminar participants at UTS, Cambridge JudgeBusiness School, City University of London, the 1st Workshop on Advances in Industrial Organization in Bergamo,the 2017 Asia-Pacific Industrial Organisation Conference in Auckland and the 2018 EARIE conference in Athensfor helpful comments and suggestions.†University of Bayreuth, CESifo, and CEPR (fabian.herweg@uni-bayreuth.de).‡University of Technology Sydney and CSEF (Antonio.Rosato@uts.edu.au).

1 Introduction

Many consumers are familiar with so-called bait-and-switch strategies whereby customers are

first “baited”by merchants’advertising products or services at a low price, but upon visiting the

store, are then pressured by sales people to consider similar, but more expensive, items (“switch-

ing”). In a series of articles appeared in the “The Haggler”– a column in the Sunday edition

of The New York Times (NYT) – journalist David Segal describes a somewhat different strategy

employed by large retailers like Staples, BestBuy and others, which he dubs bait-and-ditch: es-

corting shoppers out of the store, empty-handed, when it’s clear they have no intention of buying

an expensive warranty or some other add-on for some steeply discounted electronic appliance –

a practice that employees at Staples themselves call “walking the customer.”He further reports

that clerks and sale representatives, at Staples and elsewhere, are under enormous pressure to sell

warranties and accessories, particularly on computers. For motivation, close tabs are kept on the

amount of extras and service plans sold for each and every computer; the goal is to sell an average

of $200 worth of add-ons per machine, and a sales clerk who cannot achieve the goal is at risk of ter-

mination. Therefore, sale representatives prefer to forgo the sale altogether, rather than selling the

base good without the add-on. (“Selling It With Extras, or Not at All”: http://nyti.ms/PcobqM).

In the last few years there has been a dramatic increase in the sale of add-on products such

as extended warranties, service plans, credit insurance and financing programs like “buy now, pay

later” arrangements, and it is well known that such practices generate a substantial fraction of

retailers’ profits in different consumer industries. Likewise, the use of sales quota to motivate

sales representatives is not novel nor is the fact that meeting one’s quota is usually an attractive

goal as it leads to additional benefits such as promotion or job security (see Oyer, 2000). Yet,

the NYT article highlights how retail chains design compensation schemes that push their sales

people to target and exploit naïve or less savvy consumers, concluding that such compensation

schemes might backfire in the end.1 Indeed, it is well known that often sales people successfully

“game”incentive systems by taking actions that increase their pay but hurt the objectives of their

employer, such as manipulating prices, influencing the timing of customer purchases, and varying

effort over their firms’fiscal years.2

In this paper, we look at the interplay between employees’compensation schemes and market

competition when (some) consumers are naïve. We start from the same premise as the NYT

1In the context of financial advice, Anagol, Cole, and Sarkar (2017) report evidence that suggests that salesagents tend to cater to, rather than correct, customers’biases. That financial advisers often reinforce customers’biases that are in their interest is also documented by Mullainathan, Noeth and Schoar (2012).

2Oyer (1998) argues that as firms often use the fiscal year as the unit of time over which many sales quota andcompensation schemes are measured, a salesperson who is under pressure to meet a quota near the end of the yearmay offer a customer a bigger price discount if the client orders immediately. In fact, he shows that firms tendto sell more (and at lower margins) near the end of fiscal years than they do in the middle of the year. Similarly,Larkin (2014) analyzes the pricing distortions that arise from the use of non-linear incentive schemes at an enterprisesoftware vendor and finds that salespeople are adept at gaming the timing of deal closure to take advantage of thevendor’s commission scheme.

1

article – that firms’attempts to exploit consumer naïveté might lead them to design somewhat

perverse incentive contracts – and show that a firm, by using these seemingly perverse incentive

contracts, is able to increase its profits. In particular, our analysis shows that it may be optimal

for a retailer to design a compensation scheme that incentivizes its salesforce to exclusively target

naïve consumers. The reason is that the contract between the retailer and its salesforce acts as

a credible commitment device ensuring that the retailer will not attempt to capture the whole

market. This, in turn, induces other retailers in the market to price less aggressively, thereby

softening price competition.

Section 2 introduces our baseline model. We analyze a model of price competition between a

transparent retailer and a deceptive one. The transparent retailer is an independent local shop

managed by its owner. The deceptive retailer is a franchise retailer which belongs to a chain

and is operated by an agent (or manager) on behalf of the chain company. The two retailers

sell an identical base product, but the deceptive retailer also offers an add-on. There is a unit

mass of consumers with heterogeneous willingness to pay for the base good. Consumers can be

either sophisticated or naïve. A sophisticated consumer understands that the add-on offered by

the deceptive retailer is worthless. A naïve consumer, on the other hand, can be convinced by

the agent that the add-on increases the value of the base good; i.e., s/he can be “talked”by the

agent into buying the add-on next to the base product. Our main contribution is to show that

by designing an appropriate compensation scheme for its manager, the chain can induce a pricing

equilibrium in which both retailers obtain “abnormal”profits (i.e., above competitive levels). The

chain can achieve this outcome by offering its store manager a contract that pushes him to never

sell the base good without the add-on. In this case, we say that the chain is engaging in “bait-

and-ditch” by inducing its manager not to serve those consumers who do not wish to buy the

add-on. Hence, complete market segmentation arises in equilibrium with the deceptive retailer

targeting only naïve consumers while the transparent retailer serves only rational ones. Market

segmentation softens price competition and eliminates the incentives for the retailers to undercut

each other’s price for the base good. Moreover, we also show that the transparent retailer might

obtain a higher profit than the deceptive one. Hence, our analysis implies that even transparent

firms may strictly prefer not to educate naïve consumers, even when education is costless.

The idea that contractual delegation to an agent can be profitable for firms’owners is not new.

Several authors (e.g., Fershtman, 1985; Vickers, 1985; Fershtman and Judd, 1987; Sklivas, 1987)

have shown how, by using an appropriate incentive contract, a firm can commit to behave more (or

less) aggressively than it would without delegation. In particular, a firmmay use seemingly perverse

incentive schemes. For instance, Fershtman and Judd (1987) showed how owners can benefit by

inducing their managers to keep sales low.3 Our paper differs from these previous contributions in

the assumed market structure and, more importantly, in how we model consumer behavior. The

different underlying assumptions generate novel results and implications. For example, our model

3Delegation can also be used to collude more effectively; see Fershtman, Judd and Kalai (1991) and Lee (2010).

2

predicts price dispersion in the based good’s price despite the fact that retailers supply identical

products and have the same costs. Classical models of price dispersion (e.g., Salop and Stiglitz,

1977; Varian, 1980) rely on the presence of significant search costs for consumers and on price

randomization on the part of firms. In our model, instead, consumers are all perfectly informed

about the price(s) of the base good and the pricing game’s equilibrium is in pure strategies.

Nonetheless, price dispersion arises as a by-product of the endogenous market segmentation.

The finding that the chain can increase its profits by committing to sell only the bundle is —

at first glance —reminiscent of the leverage theory of tied sales (Whinston, 1990). According to

this theory, bundling is beneficial for the firm that offers both products but hurts the profits of the

competitor who offers only one product. By contrast, in our model both firms obtain higher profits

if the chain commits to sell the base good only together with the add on. Therefore, our finding

that the deceptive retailer can increase its profits by committing to serve only naive consumers,

who buy the add-on together with the base good, is more similar to the role of “technological

bundling” as a tool to relax price competition. Chen (1997) considers a duopoly model where

firms can commit to sell only the bundle via technological bundling. In equilibrium, one firm uses

pure bundling while the other specializes by offering only one product. Our model, however, differs

on two crucial aspects. First, naïve consumers are attracted by the bundle only if it is cheaper than

the base good offered at the competing retailer. Second, the ability to commit to offer only the

pure bundle is intrinsically connected to the fact that the deceptive retailer uses a compensation

scheme that induces its agent to target only naïve consumers.

Our model also delivers interesting policy implications. First, we find that welfare is not

monotone in the fraction of naïve consumers in the market, implying that policy interventions

designed to de-bias naïve consumers can backfire. The reason is that market segmentation is not

an equilibrium if the fraction of naïve consumers is relatively high, as in this case the transparent

retailer would not be content with serving only sophisticated consumers; hence, a reducing the frac-

tion of naïve consumers may actually help firms to achieve perfect market segmentation. Johnen

(2017) obtains a similar result whereby firms’profits and consumer surplus are non-monotone in

the fraction of naïve consumers. In his model, a customer base with more sophisticates induces

more severe adverse selection of unprofitable sophisticates and less-intense poaching, thereby soft-

ening competition. Second, we show that welfare is higher if the deceptive retailer is a monopolist

compared than if it competes with a transparent retailer. Indeed, a monopolistic deceptive retailer

has an incentive to serve both naïve as well as sophisticated consumers. As a result, compared to

the bait-and-ditch outcome under duopoly, a larger fraction of sophisticated consumers buy the

base good (at a lower price) and a lower fraction of naïve consumers buy the bundle (at a higher

price). As the additional sophisticated consumers who are served under monopoly have a higher

willingness to pay than the naïve ones that are served only under duopoly, consumer and total

welfare is higher under monopoly. Hence, fostering competition by adding a transparent firm in a

market where a deceptive firm operates as a monopolist might be welfare detrimental.

3

Section 3 analyzes two extensions of our basic framework. In Section 3.1 we enrich the baseline

model to allow the manager of the deceptive retailer to exert private effort that enhances the

probability that a customer buys the add-on. Moreover, the manager also incurs a cost when

walking out consumers who are not willing to purchase the add-on. Intuitively, the addition of

this two-task agency problem makes it more costly for the deceptive retailer to engage in bait-and-

ditch. Nevertheless, we show that it is often profitable for the deceptive retailer to do so even if it

has to pay an information rent to its agent. Interestingly, we also identify a “complementarity of

ineffi ciencies”whereby the chain company is more likely to induce the manager to exert (socially

costly) effort when it engages in bait-and-ditch.4 Section 3.2 extends our baseline model beyond

the case of duopoly by introducing a competitive fringe that supplies an imperfect substitute for

the base good supplied by the deceptive and transparent retailers, and shows that bait-and-ditch

still arises in equilibrium if these retailers retain suffi cient market power.

Section 4 concludes the paper and discusses possible avenues for future research. The remainder

of this section discusses the literature most closely related to our paper.

Our paper joins the recent literature on consumer naïveté. Starting with the seminal contri-

bution of Gabaix and Laibson (2006), most papers in this literature focus on the incentives (or

lack thereof) for deceptive firms to educate naïve consumers by unshrouding their hidden fees or

attributes, and derive conditions for a deceptive equilibrium – one in which naïve consumers are

exploited – to exist.5 The main implications of this literature are twofold: (i) deceptive firms

do not want to educate/de-bias naïve consumers as this would turn them from profitable into

unprofitable; and (ii) the presence of naïve consumers benefits rational ones who take advantage

of low-priced base goods (often loss leaders) but do not buy the expensive add-ons. While related,

our paper differs from previous contributions in this literature on several key dimensions. First,

we do not focus on the question of whether firms want to educate consumers as we consider an

asymmetric set-up with one deceptive firm and one transparent firm (or more). Nevertheless,

we find that even in such an asymmetric environment, a deceptive equilibrium can be sustained.

Moreover, in most of the models in this literature, deceptive and transparent equilibria result in

the same profits for the firms as profits gained from naïve consumers via the add-on are passed

on to sophisticated consumers via a lower price on the base good; in our model, instead, both the

deceptive firm and the transparent one attain strictly higher profits in a deceptive equilibrium,

with the transparent firm potentially obtaining the lion’s share of the total profits. Furthermore,

in our model the presence of naïve consumers actually hurts sophisticated ones.6 The reason is

4Inderst and Ottaviani (2009) analyze a two-task agency model of sales where an agent needs to prospect forcustomers as well as advise them on the product’s suitability. In their model the two tasks are in direct conflictwith each other so that when structuring its salesforce compensation, a firm must trade off the expected losses from“misselling”unsuitable products with the agency costs of providing marketing incentives to its agent. In our model,instead, in equilibrium the agent’s tasks end up being complementary with one another.

5See also Armstrong and Vickers (2012), Dahremöller (2013), Miao (2010), Murooka (2015), Heidhues, Koszegiand Murooka (2016, 2017) and Johnen (2017).

6Analyzing a competitive insurance market with rational and overconfident consumers, Sandroni and Squintani

4

that in equilibrium naïve and sophisticated consumers buy from different retailers and this relaxes

price competition on the base good. Indeed, in our model sophisticated consumers end up buying

the base good at the monopoly price (if they buy at all). Within this literature, the papers most

related to ours are Heidhues and Koszegi (2017), Kosfeld and Schüwer (2017), and Michel (2017).

Like ours, all these models show that firms can increase their profits thanks to the ability to target

naïve consumers. Yet, Heidhues and Koszegi (2017) and Kosfeld and Schüwer (2017) are models

of third-degree price discrimination where firms can condition the terms of their offers on external

information about consumers’naïveté. Our model, instead, is one of uniform pricing where retail-

ers only know that a fraction of consumers are naïve, but cannot tell ex-ante which consumers

are naïve and which ones are not. Similar to us, Kosfeld and Schüwer (2017) find that educating

consumers may backfire as in their model a larger share of sophisticated consumers may trigger

an equilibrium reaction by firms that is undesirable for all consumers. Michel (2017) is more in

the vein of Heidhues, Koszegi and Murooka (2016). He explicitly models extended warranties as

useless add-on products. As in our model, naïve consumers do not pay attention to the add-on

when choosing which store to visit, but then overestimate the add-on’s value at the point of sale. In

contrast to us, Michel (2017) analyzes a symmetric game between firms and is primarily concerned

about the welfare effects of consumer protection policies; e.g. a minimum warranty standard.

Our paper is also related to the literatures on bait-and-switch, add-on pricing and loss leaders.

Lazear (1995) studies a duopoly model with differentiated goods and derives the conditions for

bait-and-switch to be profitable; in his model bait-and-switch is purely false advertising as a

firm claims to sell a different good than the one it actually produces. Hess and Gerstner (1987)

develop a model where firms sometimes stock out on advertised products and offer rain checks

because consumers buy “impulse goods”whenever they visit a store to buy an advertised product.

Gerstner and Hess (1990) present a model of bait-and-switch in which retailers advertise only

selected brands, low-priced advertised brands are understocked and in-store promotions are biased

towards more expensive substitute brands. Balachander and Farquhar (1994) show that by having

occasional stockouts firms can soften price competition and hence “gain more by stocking less.”

Lal and Matutes (1994) develop a model of loss-leader pricing in which every consumer purchases

the same bundle at the same price regardless of whether the prices of add-ons are advertised or

not. Verboven (1999) analyzes a model of add-on pricing where consumers differ in their marginal

willingness to pay for quality and shows that add-on pricing again has no effect on profits. Ellison

(2005) proposes a price-discrimination model in which add-on pricing enables firms to charge high-

demand consumers relatively more than low-demand consumers. In his model, search costs make it

costly for consumers to observe add-on prices and high add-on markups raise profits by facilitating

price discrimination. Finally, Rosato (2016) proposes a model of bait-and-switch in which a retailer

offers limited-availability bargains to exploit consumers’loss aversion.

(2007) show that the presence of overconfident (naïve) consumers can hurt rational ones.

5

2 Baseline Model

Consider a market with two retailers, denoted by D and T . Retailer T is an independent

local shop managed by its owner. Retailer D is a franchise retailer that belongs to a chain. Both

retailers offer the same base product; e.g. a laptop. The prices charged by retailer D and T for

the base good are denoted by pD and pT , respectively. For simplicity we assume that the costs for

the base good (wholesale prices) are zero for both retailers. Retailer D can also offer an add-on;

e.g. an extended warranty or service plan.7 The price for the add-on is fD and selling the add-on

is without costs for retailer D. Retailer T , on the other hand, has prohibitively high costs for

offering the add-on and therefore offers only the base good.8

There is a unit mass of consumers interested in purchasing one unit of the base good. A

consumer’s willingness to pay for the base good is denoted by v. We assume v is distributed

uniformly on the unit interval; i.e., v ∼ U [0, 1]. Each consumer can be either sophisticated or

naïve and the probability of being naïve, which is independent of the willingness to pay, is denoted

by σ ∈ (0, 1). The add-on is worthless and a sophisticated consumer understands this. In other

words, owning the add-on does not increase a sophisticated consumer’s willingness to pay for the

base-good. A naïve consumer, on the other hand, can be convinced by a sales agent that the

add-on increases the value of the base good by f ∈ (0, 1); i.e., a naïve consumer can be “talked”

into paying up to f for the add-on if he purchases the base-good as well.9

Retailer T is operated by its owner, who chooses the price for the base good in order to maximize

the shop’s profit πT . Retailer D is operated by a sales agent who reacts on the incentive payments

offered to him by the chain. When the agent’s compensation depends solely on the profits made

by the retail outlet, πD, he chooses the base good and the add-on prices to maximize πD. The

chain company could also offer a more complex compensation scheme depending, for instance,

on the revenue generated by the add-on sales. Why offering such a scheme, which might create

misalignment between the chain company’s and the manager’s interests, can be optimal will be

explained later. We posit that the store manager has an outside option yielding a utility of U = 0.

The sequence of events is as follows:

1. Retailer D offers a contract to its sales agent, who either accepts or rejects the offer. The

agent’s decision as well as the terms of the contract are observed by T .

7This asymmetric market structure can be interpreted as the final stage of a two-stage game where in the firststage ex-ante symmetric firms simultaneously choose their organizational structure; e.g., whether to hire an agentand supply the add-on. It is easy to see then that in any (pure-strategy) subgame-perfect equilibrium the two firmswill select different organizational structures in the first stage, as otherwise they would both obtain zero profits inthe last stage.

8This is consistent with the observation that many large consumer electronics retailers offer their own extendedwarranties whereas smaller shops are usually not able to do so; see, for example, OFT (2012) for recent evidence.

9Hence, we interpret the add-on offered by the deceptive retailer as a purely worthless product that naïveconsumers can be tricked into buying; see Armstrong (2015) for a similar model. For a richer model of “sales talk”with rational and credulous consumers, see Inderst and Ottaviani (2013).

6

2. The sales agent of D and the owner of T simultaneously set prices for their products.

3. Consumers with a willingness to pay v ≥ min{pD, pT} visit the cheaper retailer first. If T ischeaper, all these consumers —sophisticates and naïfs —purchase the base good from T . If

D is cheaper, the agent decides whether to sell only the bundle – base good + add-on – at

price pD + fD or to sell also the base good by itself at price pD. In the former case, we say

that sophisticated consumers are walked out of the shop. If pD = pT , we assume consumers

visit retailer D first.10

The analysis is decomposed into two parts. First, we analyze the equilibrium of the pricing

game for the case where the agent of D cares only about the profits of the retail outlet he manages.

This situation is equivalent to the chain company having full control over the strategic decisions

of retailer D. Thereafter, we assume that the agent has an incentive not to serve sophisticated

consumers. We derive the equilibrium of the pricing game under this presumption and then obtain

suffi cient conditions so that the agent indeed does not want to serve sophisticates. This second

scenario can also be thought of as delegation —the chain company delegates all strategic decisions

to the sales agent. Finally, we compare the two scenarios and show that bait-and-ditch may occur

in a subgame perfect equilibrium.

2.1 Manager maximizes profits

Suppose that the compensation of the agent of D depends positively on the profits of the chain

and on nothing else. Then, both the agent of D and the owner of T choose prices to maximize the

profits of their respective stores. Hence, there is Bertrand competition for the base good and the

equilibrium prices are

pD = pT = 0. (1)

The add-on is offered only by D and thus charging the monopoly price for it is optimal; i.e.,

fD = f . (2)

The profits of the retailers are

πD = σf πT = 0. (3)

Sophisticated consumers are indifferent between the two retailers and the equilibrium outcome

is independent on how we break the indifference.11 In order to achieve this outcome, the chain

10This tie-breaking rule is assumed only for expositional simplicity and to guarantee equilibrium existence. Ifanything, this assumption favors retailer D and thus reduces the incentives for the chain to use the “commitmentstrategy”of not serving sophisticated consumers.11In order to see why this is the unique equilibrium outcome, suppose D were to serve only naïve consumers; i.e,

it does not serve sophisticates. Now, retailer T can demand a strictly positive price from the unserved sophisticates.This, however, cannot be an equilibrium. Retailer D is not committed to serve only naïfs and therefore has an

7

company could offer the following wage contract to its sales agent who manages retailer D

w(πD) = πD − σf . (4)

The agent accepts this contract and all rents accrue to the chain company whose profit is

Π = σf . (5)

In essence, the chain company charges the manager of store D a franchise fee equal to σf .

2.2 Walking sophisticated consumers

Suppose now that the manager of retailer D has an incentive not to serve sophisticated con-

sumers; i.e., to walk sophisticated consumers out of the store. In the following we characterize an

equilibrium of the pricing game in which retailers make “abnormal”profits. We posit that the D

is committed to serve only naïve consumers and T is aware of this commitment. Thereafter, we

investigate whether this commitment can be achieved by an appropriate incentive scheme offered

by the chain company to the manger of D.

Assume there is an equilibrium in which D serves only naïve consumers and T serves only

sophisticated consumers. If such an equilibrium exists then for any price pD of the base good that

D charges, it is optimal for this retailer to set the price of the add-on good at its highest possible

level; i.e., fD = f . In the Appendix we formally establish this result by considering deviations on

both prices, pD and fD, simultaneously. The price pi, with i = D,T, has to maximize the retail

profit πi under the presumed market segmentation. The profit of retailer D is

πD(pD) = σ(1− pD)(pD + f). (6)

All naïve consumers with a willingness to pay of v ≥ pD purchase from retailer D. Each naïve

consumer purchases next to the base good also the add-on and thus each sale is worth pD + f to

the retailer. From the first-order condition we obtain

pD =1

2

(1− f

). (7)

The corresponding profit of retailer D is

πD := πD(pD) = σ[1− 1

2(1− f)

] [1

2(1− f) + f

]=

σ

4(1 + f)2. (8)

incentive to undercut the base product price of T and to serve both consumer groups. Hence, regarding the baseproduct’s price, the standard Bertrand outcome is the unique equilibrium outcome.

8

The profit of retailer T is

πT (pT ) = (1− σ)(1− pT )pT . (9)

All sophisticated consumer with a willingness to pay of v ≥ pT purchase from retailer T . They

buy only the base good at price pT . From the first-order condition we obtain

pT =1

2. (10)

The corresponding profit of retailer T is

πT := πT (pT ) =1− σ

4. (11)

Hence, we have the following result:

Proposition 1. Suppose retailer D is committed not to serve sophisticated consumers. Then, if

σ ≤ f 2 there exists a Nash equilibrium of the pricing game with higher than Bertrand profits. The

equilibrium prices and profits are

pD =1− f

2, pT =

1

2, fD = f , πD =

σ

4(1 + f)2, πT =

1− σ4

.

If retailer D is able to commit not to serve sophisticated consumers, and if the fraction of

naïve consumers in the market is not too high, both retailers are able to achieve higher than

Bertrand profits. The intuition for this result is that when retailer D is committed to serve only

naïve consumers, complete market segmentation arises in equilibrium with firm T targeting only

sophisticated consumers while firm D targets only naïve ones. Hence, the two retailers essentially

operate as “local”monopolists. Market segmentation, in turn, softens price competition on the

base good so that both retailers are able to charge prices above marginal cost. Furthermore,

it is worth highlighting that the Nash equilibrium of the pricing game described in Proposition 1

features price dispersion in the based good’s price despite the fact that the retailers supply identical

products and have the same costs. Indeed, it is easy to verify that

pD < pT < pD + f . (12)

Intuitively, retailer D has a direct incentive to lower the price of the base good to sell more

units as it can extract more per-sale profits via the add-on. This, in principle, would induce retailer

T to match (or undercut) retailer D until all profits from the base good are competed away. Yet,

if retailer D is committed to serve only naïve consumers, retailer T can charge the monopoly

price for the base good and extract the monopoly profit from sophisticated consumers. Extracting

monopoly rents from sophisticated consumers leads to a higher profit than slightly undercutting

retailer D if the share of sophisticates is suffi ciently high, i.e., if σ ≤ f 2. Finally, notice that while

9

both retailers achieve strictly positive profits in the equilibrium described in Proposition 1, it is

not necessarily the case that retailer D is the one benefiting more in this equilibrium. Indeed, it

is easy to verify that

πT ≥ πD ⇐⇒ σ ≤ 1

1 + (1 + f)2. (13)

Therefore, if the fraction of naïve consumers is low enough, retailer T attains higher profits

than retailer D, despite the fact that retailer T does not sell an add-on product. This, in turn,

generates the novel, interesting implication that even if a firm is not the one obtaining the largest

profit in the market, it may still be possible that the firm is serving its product deceptively, thereby

exploiting naïve consumers.

2.3 Optimal strategy and corresponding sales contract

The question at hand now is: How can the chain company, in the first stage of the game,

achieve that its agent does not serve all customers that are willing to pay the base good’s price? In

other words, how can retailer D credibly commit not to serve sophisticated consumers? The chain

company can offer its agent a wage payment w = w(rB, rA), which depends both on the base-good

revenue, rB, and the add-on revenue, rA, generated by the store. Hence, in order to achieve the

outcome of the pricing subgame equilibrium described in Proposition 1, the chain company could

offer its agent the following compensation scheme:

w(rB, rA) = min {rA, rA}+ min {rB, rB} − F, (14)

where rB = σ4(1 − f 2), rA = σ

2(1 + f)f and F = rB + rA is a fixed franchise fee. For this

compensation scheme, it is readily established that an optimal strategy for the agent is to set

pD = pD, fD = fD, and not to serve sophisticated consumers. There are two crucial elements in

this compensation scheme. First, according to this contract, the agent gets to keep the revenues

from both add-on and base-good sales up to the target values rB and rA. Therefore, there is no

incentive for the agent to try to increase the store revenue beyond rA + rB. The second crucial

aspect of this compensation scheme is that it specifies two distinct revenue targets: one for the

sales of the base good and one for the add-on sales. If the chain were to specify a target for overall

revenue, instead, this would not work as a credible commitment device not to serve sophisticated

consumers because it would give the sales agent too much leeway in choosing prices and re-shuffl ing

revenue between sales of the base good and add-on sales. In turn, then, retailer T would price its

base good more aggressively in an attempt to gain further market share. Moreover, notice that,

as all relevant variables are observable, there is symmetric information between the chain and the

agent. Hence, by choosing F appropriately, the chain company can acquire all the rents in the

10

end. The chain’s profit —when offering the incentive scheme (14) —is

Π =σ

4(1 + f)2. (15)

The following proposition summarizes this result:12

Proposition 2. Suppose that σ ≤ f 2. Then there exists a subgame perfect equilibrium where the

chain offers its agent the compensation scheme in (14) so that he has no incentives to sell the base

good without the add-on; i.e., the sales agent walks sophisticated customers out of the store.

Hence, by steering the incentives of its agent away from simple (downstream) profit maximiza-

tion, and providing him instead with an incentive not to serve sophisticated consumers, the chain

company is able to sustain an equilibrium with “abnormal” profits. The point that consumer

naïveté can generate “abnormal”profits in markets with add-on pricing or deceptive products has

already been recognized by several authors, including Ellison (2005), Gabaix and Laibson (2006),

Armstrong (2015) and Heidhues, Murooka and Koszegi (2016, 2017). Yet, in these models the

firms are symmetric and can all offer an add-on or deceptive product. In our model, instead, the

firms are extremely asymmetric on this dimension, with only one firm being able to sell an add-on;

yet, an “exploitative”equilibrium is still possible.13

Another difference with respect to the prior literature on consumer naïveté is that in our model

the presence of naïve consumers imposes a negative externality on rational ones, whereas in most of

the models mentioned above rational consumers benefit from the presence of naïve ones. Indeed, in

our model each type of consumer, rational or naïve, would be better off if they were the only type

in the market. Moreover, notice that the equilibrium described in Proposition 2 is highly ineffi cient

because a positive measure of naïve as well as sophisticated consumers end up being priced out of

the market for the base good. Specifically, all sophisticated consumers with v < 12and all naïve

consumers with v < 1−f2end up not buying the base good. Hence, in addition to redistributing

surplus away from the consumers and towards the firms, the practice of bait-and-ditch also lowers

total welfare in the market. Indeed, as the following proposition shows, welfare would be higher

with a single retail outlet operated by D:

12For the simple wage scheme (14) the equilibrium outcome is not unique. In particular, pD = pT = 0 is also partof a subgame perfect equilibrium. For more complex wage schemes the outcome described in Proposition 2 is theunique equilibrium outcome. In order to see this, suppose the manager obtains a sizable bonus for each add-on salebut is mildly punished for each sale of the base good. Under such a scheme the manager has a strict incentive tosell the base good only together with the add-on. This is known by retailer T who now has an incentive to chargea strictly higher price on the base good than retailer D, with pD < pT < pD + fD. Hence, zero base good pricesare no longer part of a subgame perfect equilibrium.13A notable exception is the model presented by Murooka (2015) where a deceptive and a non-deceptive firm sell

to consumers via a common intermediary. He shows that a deceptive equilibrium, whereby only the deceptive firmends up selling to consumers, is possible despite the firms being asymmetric. Yet, the intuition behind his resultdoes not hinge on market segmentation as a mechanism, but rather on the common agency of the intermediary.Moreover, in his model all consumers are naïve ex-ante (but can be “educated”by the intermediary).

11

Proposition 3. Suppose that σ ≤ f 2. Then total welfare is higher when the deceptive retailer is

a monopolist than when it competes with a transparent retailer.

Proposition 3 points out that fostering competition in a market where a deceptive firm operates

as a monopolist might actually be welfare detrimental. While it may appear counterintuitive at

first, the intuition behind this result is as follows. First, notice that if it were a monopolist, retailer

D would charge pmD = 1−σf2for the base good and fmD = f for the add-on. Hence, the overall fraction

of consumers that end up buying the base good is the same. However, compared to the bait-and-

ditch outcome under duopoly, a larger fraction of sophisticated consumers would consume the base

good (at a lower price) whereas a lower fraction of naïve consumers would consume the bundle (at

a higher price). As the additional sophisticated consumers who are served under monopoly have

a higher willingness to pay than the naïve ones that are served only under duopoly, consumer and

total welfare is higher under monopoly.

What measures should a social planner undertake to increase welfare? Interestingly, an im-

portant implication of our model is that social welfare is non-monotone in the fraction of naïve

consumers. Therefore, consumer education policies aimed at increasing consumer sophistication

in the market might be counterproductive and welfare detrimental.14 Indeed, let σ1 denote the

initial fraction of naïve consumers in the market and suppose σ1 > f 2. In this case, the Bertrand

outcome (for the base good) is the unique equilibrium of the pricing game. It is true that in this

equilibrium naïve consumers are taken advantage of since they end up buying a worthless add-on

and paying f for it. Yet, a policy that reduces the fraction of naïve consumers in the market

can hurt both naïve as well as sophisticated consumers. To see why, let σ2 < σ1 denote the new

fraction of naïve consumers after the policy intervention. Unless σ2 = 0, the effect of the policy is

ambiguous ex-ante. If σ2 > f 2, the Bertrand outcome continues to be the only equilibrium, but

now the fraction of exploited consumers is reduced; in this case, the policy increases welfare. Yet,

if σ2 < f 2, then the policy is giving the deceptive retailer exactly the commitment power neces-

sary to engage in bait-and-ditch and achieve perfect market segmentation. On the other hand,

mandating retailers to issue rainchecks when advertised products are (claimed to be) out of stock

would unambiguously improve consumer and total welfare.

Can the chain company achieve the bait-and-ditch outcome by (technological) bundling the

base good with the add-on good instead of relying on an incentive sales contract? The answer

is no. Indeed, naïve consumers would still visit the retailer with the lower total price. Hence,

retailer D cannot charge a higher price for the bundle than the one retailer T charges for the base

product alone. The usual Bertrand outcome then arises as competition is not relaxed. Importantly,

retailer D first has to attract naïve consumers into the store with a low price on the base good,

and then it can exploit their naïveté by making them purchase the over-priced add-on. Moreover,

14Huck and Weizsäcker (2016) obtain a somewhat similar implication in markets for sensitive personal informationwhere a fraction of consumer is naïve and underestimate the chance that their private information will be revealedto a third party.

12

the Bertrand outcome can be avoided only if the manger of retailer D does not want to serve the

sophisticated consumers who might also be attracted to the store by the low price on the base

good. In other words, the agent has to take two important (endogenous) actions: (i) talking naïve

consumers into buying the add-on, and (ii) walking out sophisticated consumers empty-handed.

While these actions are fairly abstract in our baseline model, in the extension that we discuss in

the next section, these actions have to be explicitly incentivized by the chain.

Finally, we point out that there exists an equivalent “standard”version of our model where

all consumers are rational; i.e., where naïve consumers have a true willingness to pay for the

add-on equal to f . The same type of equilibria would exist also under this alternative “rational”

interpretation. However, the welfare implications are slightly different because consumption of

the add-on good is now desirable from a welfare perspective. Nevertheless, market segmentation

still results in prices for the base good that are too high and hence detrimental for welfare. Most

importantly, under this alternative “rational” interpretation — in contrast to our interpretation

where some consumers are naïve —bundling of the two goods is a feasible strategy for the chain

to achieve commitment. Hence, in this case there would be no role for the sales agent.

3 Extensions and Robustness

In order to clearly highlight the key intuition behind our results, in the baseline model we

assumed a duopolistic market structure and symmetric information between the deceptive retailer

and its sales agent. In this section we separately relax each of these assumptions. Section 3.1

extends our baseline model by allowing the sales agent to privately exert effort in order to enhance

the probability that a customer is willing to purchase the add-on good. Hence, the chain now

has to offer a contract that mitigates the moral-hazard problem. Section 3.2 extends the baseline

model of Section 2 to the case where there are more than two retailers supplying the base good.

3.1 Model with moral hazard

There is a mass one of consumers with willingness to pay for the base good equal to v, which is

uniformly distributed on the unit interval; i.e., v ∼ U [0, 1]. A fraction of these consumers is naïve

and willing to purchase the add-on good as long as its price is not larger than f . The fraction of

naïve consumers is σ ∈ {L,H}, with 0 < L < H < 1. Crucially, the fraction of naïve consumers

is stochastic and the probability distribution over σ can be affected by the agent’s effort. If the

agent works hard, his sales talk is more convincing and thus it is more likely that a customer will

buy the add-on. We model this in the following simple way. The agent can choose a binary effort

level e ∈ {0, 1}. The cost of effort is given by ψe, with ψ > 0. The probability that a large fraction

13

of customers is naïve depends on the effort level e and is given by

Pr(σ = H| e) = qe.

We impose the following standard full-support assumption:

Assumption 1. 0 < q0 < q1 < 1.

Let σe = qeH + (1− qe)L denote the expected fraction of naïve consumers conditional on theagent’s effort e ∈ {0, 1}. Notice that by Assumption 1 σ0 < σ1. We assume that the fraction

of naïve consumers is always relatively low —independent of the effort choice —compared to the

willingness to pay for the add-on:

Assumption 2. σ1 < f 2.

Moreover, the agent also has to suffer a fixed cost φ > 0 in order to “walk out”those consumers

who do not wish to buy the add-on.15 For simplicity we assume that this cost is fixed and does

not depend on the number of customers that are walked out empty-handed. Moreover, we assume

that it is relatively more costly for the agent to talk consumers into buying the add-on than to

“walk out”those consumers who do not wish to buy it:

Assumption 3. φ ≤ ψ.

The chain company —and also a third party —can verify the number of base goods and add-ons

sold at the store. It also knows the prices it charges for both goods. In other words, ex post the

chain company observes the state of the world σ and whether the agent has served both groups of

consumers or only the naïve ones. The agent’s remuneration, i.e. his wage, can be contingent on

all these variables. Thus, the chain specifies four wage payments:

w = {wL,N , wL,S, wH,N , wH,S},

where wσ,j denotes the wage paid by the chain and received by the agent if the state is σ and if

consumer group j ∈ {N,S} is served. Here, j = N denotes the case when only naïve consumers

are served while j = S denotes the case when both naïve and sophisticated consumers are served.

Finally, we assume that the agent is risk neutral but protected by limited liability and thus

cannot make any payments to the chain; i.e., the limited-liability constraint,

wσ,j ≥ 0 ∀ σ ∈ {L,H}, j ∈ {N,S},

needs to be satisfied. The sequence of events is as follows:15This could be interpreted either as the physical cost of having to go through the whole charade of going into the

back of the store and pretend to check whether there is any unit of the base-good still availabe; or, alternatively, ifone thinks that the agent is intrinsically sympathetic towards the consumers, as the psychological cost of having tolie to the consumers and letting them go empty-handed.

14

1. Contracting stage: the chain offers a wage contract w to its agent, who either accepts or

rejects the offer. The terms of the contract offered by the chain as well as the agent’s decision

to accept or reject the contract are publicly observed.

2. Pricing stage: the chain and the local retailer T simultaneously set prices.

3. Effort stage: If the agent accepted the contract, he chooses an effort level e and decides

whether or not to sell only the bundle, base good plus add-on; i.e., whether to engage in

bait-and-ditch.

4. Purchasing stage: consumers decide whether and where to buy.

3.1.1 Serving both consumer groups

Suppose the chain wants its agent to serve both types of consumers with the base good and

additionally to sell the add-on to naïve consumers. Irrespective of the anticipated effort choice

of the agent, there is Bertrand competition for the base good and thus the equilibrium prices are

pD = pT = 0. Retailer D is a monopolist for the add-on good and thus fD = f . In this case, the

chain solves the following maximization program:

maxe,wσ,j

σef − qewH,S − (1− qe)wL,S (S)

subject to: for e ∈ {0, 1} and e 6= e

qewH,S + (1− qe)wL,S − eψ ≥ 0 (PCSe )

qewH,S + (1− qe)wL,S − eψ ≥ qewH,S + (1− qe)wL,S − eψ (ICSe )

qewH,S + (1− qe)wL,S − eψ ≥ qewH,N + (1− qe)wL,N − eψ − φ (ICSS)

qewH,S + (1− qe)wL,S − eψ ≥ qewH,N + (1− qe)wL,N − eψ − φ (ICSe,S)

wσ,j ≥ 0 for all σ ∈ {L,H}, j ∈ {N,S} (LL)

The chain maximizes its expected profit — given that it prefers to serve both sophisticated

and naïve consumers —subject to five constraints. First, the agent has to accept the offer; i.e.

(PCSe ) has to hold. Next, the agent has to choose the intended level of effort, constraint (ICSe ),

and has to serve both consumer groups, constraint (ICSS). The agent can also jointly deviate; i.e.,

choosing the wrong effort level and serving only naïve consumers. Therefore, the joint incentive

constraint (ICSe,S) needs to be satisfied. Finally, due to limited liability (LL), the chain has to

specify non-negative wages.

The agent has no intrinsic motivation to serve only naïve consumers. This implies that if the

chain wants that the agent serves both types, no incentive payment is necessary to achieve this,

i.e., the constraints (ICSS) and (ICSe,S) do not impose a binding restriction.

15

Thus, if the chain wants to induce low effort, e = 0, it is straightforward to show that it is

optimal to pay a zero wage in all states, wσ,j = 0 for all σ, j. The corresponding profit of the chain

is

Π0 = σ0f .

Next, suppose the chain prefers that the agents works hard; i.e., e = 1. If the limited liability

(LL) and the effort incentive constraint (ICS1 ) are satisfied, then also participation (PCS1 ) holds.

The optimal wage scheme is:

wH,S =ψ

q1 − q0

, wL,S = wH,N = wL,N = 0.

The chain’s expected profit amounts to

Π1 = σ1f −q1

q1 − q0

ψ.

Thus, if the chain wants to serve both types of consumers, it induces its agent to exert high

effort if and only if

ψ ≤ (H − L)(q1 − q0)2f

q1

=: ψ.

3.1.2 The chain serves only naïve consumers

Suppose now the chain can credibly commit to serve only those consumers who purchase the

add-on good in addition to the base good; i.e., only naïve consumers. If this is the case, the price

of the add-on is fD = f and the prices of the base good are pD = (1 − f)/2 and pT = 1/2. The

chain solves the following maximization program:

maxe,wσ,j

σe(1 + f

)2

4− qewH,N − (1− qe)wL,N (N)

subject to: for e ∈ {0, 1} and e 6= e

qewH,N + (1− qe)wL,N − eψ − φ ≥ 0 (PCNe )

qewH,N + (1− qe)wL,N − eψ − φ ≥ qewH,N + (1− qe)wL,N − eψ − φ (ICNe )

qewH,N + (1− qe)wL,N − eψ − φ ≥ qewH,S + (1− qe)wL,S − eψ (ICNN)

qewH,N + (1− qe)wL,N − eψ − φ ≥ qewH,S + (1− qe)wL,S − eψ (ICNe,N)

wσ,j ≥ 0 for all σ ∈ {L,H}, j ∈ {N,S} (LL)

The difference with respect to the previous program is that the chain now has to insure that

the agent does not serve sophisticates who are only interested in purchasing the base good. As the

16

agent experiences a disutility if he has to walk out customers empty-handed, the problem now is

more intricate.

Suppose the chain wants to induce low effort, e = 0, and thus does not need to specify an

incentive payment which motivates the agent to work hard. Importantly, the chain can perfectly

monitor the fraction of add-on sales over total sales and thus observes whether or not sophisticates

are served. This implies that there is no moral hazard problem between the chain and the agent

regarding the served types of consumers. The chain simply has to compensate the agent for the

disutility associated with walking out sophisticated consumers. The optimal wages are wH,S =

wL,S = 0 and wH,N = wL,N = φ. The chain’s corresponding profit is

Π0 =σ0(1 + f)2

4− φ.

Next, suppose the chain wants to induce high effort; i.e., e = 1. By the same argument as above,

it is optimal to specify wH,S = wL,S = 0. Now, satisfying the participation constraint (PCN1 ) is

suffi cient to guarantee that the agent does not want to serve also sophisticated consumers, i.e.,

the constraints (ICN1,N) and (ICNN) are redundant. As in a standard moral-hazard problem, the

effort incentive constraint (ICN1 ) will always be binding. The important question, therefore, is

whether the limited liability (LL) or the participation constraint (PCN1 ) is more restrictive. Under

Assumption 3, the limited liability constraint (LL) is binding while the participation constraint

(PCN1 ) is slack. Hence, the optimal wages are

wL,N = wH,S = wL,S = 0, wH,N =ψ

q1 − q0

.

Importantly, the expected wage cost of the chain is the same as in the case where it induces

high effort and serves both types of consumers. Intuitively, under Assumption 3, the chain’s cost

of inducing the agent to work hard on its sales talk is larger than the agent’s cost from walking

out consumers who do not want to purchase the add-on. Hence, it is suffi cient for the chain to

offer a contract that induces the agent to work hard. The disutility of the agent from walking out

sophisticated consumers does not create extra wage costs for the chain. The chain’s profit in this

case is

Π1 =σ1(1 + f)2

4− q1

q1 − q0

ψ.

If the chain serves only naïve consumers, it induces its agent to exert high effort if and only if

ψ ≤ (H − L)(q1 − q0)2

q1

(1 + f)2

4+q1 − q0

q1

φ =: ψ.

17

3.1.3 Comparison

We now investigate the overall optimal behavior of the chain; i.e., whether it prefers to serve

all consumers or to commit to serve only naïve ones. A first important observation is obtained by

comparing the critical levels of the effort cost. Indeed, it easy to verify that:

ψ > ψ. (16)

Hence, the chain is more likely to implement high effort if it is able to commit not to serve

sophisticates. From a welfare perspective low effort is preferred because effort is costly, but it

does not increase the total surplus. Moreover, the pricing equilibrium where retailer D does not

serve sophisticates is also less effi cient, from a social point of view, than the Bertrand equilibrium.

Therefore, our model features a “complementarity between ineffi ciencies” in the sense that the

chain’s gain from implementing high effort is larger when it commits not to serve sophisticates.

The chain prefers that the agent works hard if the effort cost is not too high. Importantly, the

critical effort cost is higher when the chain serves only naïve consumers than when it serves both

consumer groups. Comparing the respective profit expressions yields the next result:

Proposition 4. Suppose Assumptions 1-3 hold.

(i) For ψ ≤ ψ, the chain always offers its agent an incentive scheme such that he has no incen-

tives to sell the base good without the add-on; i.e., the agent walks sophisticated customers

out of the store.

(ii) For ψ < ψ ≤ ψ, the chain offers its agent an incentive such that he has no incentives to sell

the base good without the add-on if and only if the agent’s effort cost is not too high; i.e., if

and only if ψ ≤ q1−q0q1

σ0(1−f)2

4+ (q1−q0)2

q1

(1+f)2

4(H − L).

(iii) For ψ ≥ ψ, the chain offers its agent an incentive scheme such that he has no incentives to

sell the base good without the add-on if and only if the agent’s disutility of doing so is not

too high; i.e., if and only if φ ≤ σ0(1−f)2

4.

According to part (i) of Proposition 4, if the cost of effort is relatively low, the chain will always

induce the agent to walk out sophisticated consumers, irrespective of his disutility for doing so.

Intuitively, when the add-on good has a high profit margin, the gain from market segmentation

becomes so large that the chain never considers serving sophisticated consumers. Moreover, part

(ii) of Proposition 4 shows that, for intermediate levels of the effort cost, the decision as to whether

sophisticated consumers should be walked out is independent of the agent’s disutility for doing so.

This is a by-product of Assumption 3 which says that it is more expensive for the chain to motivate

the agent to work hard than not to serve sophisticates. Finally, in part (iii) of Proposition 4 if the

cost of walking out consumers empty-handed is not too high, the chain will design a compensation

scheme that induces its agent not to serve sophisticated consumers.

18

In this section, we have assumed that the fraction of naïve consumers who can be exploited by

retailer D directly depends on the effort exerted by its agent. Moreover, we also assumed that the

agent incurs a disutility if he has to walk out consumers empty-handed. The addition of this two-

task agency problem makes the bait-and-ditch strategy less profitable compared to the baseline

model of Section 2. Nevertheless, it is often profitable for the chain company to offer an incentive

scheme to its agent so that he engages in bait-and-ditch. Interestingly, the chain’s decision of

whether to engage in bait-and-ditch may be independent of the agent’s disutility associated with

walking out customers empty-handed. Indeed, the rent that the agent demands in order to exert

effort may already compensate him also for the disutility arising from not serving sophisticated

consumers. In this case, incentivizing the agent to serve only naïve consumers comes without

additional costs for the deceptive retailer.

3.2 More than two retailers

Our main analysis focuses on the case of duopoly. While the general question of how the

degree of competitiveness of the market affects the incentives for a deceptive retailer to engage in

bait-and-ditch is beyond the scope of the current paper, in this section we discuss a simple form of

competition between more than two retailers and show that our main result relies on some amount

of market power.

Consider a market with N ≥ 3 retailers. Retailer D sells a base product and an add-on while

retailer T sells only the base product. As in Section 2, the cost of both products are normalized

to zero and the add-on generates per-sale profits up to f . The remaining N − 2 retailers belong

to a competitive fringe and supply a base product at zero cost. There is a unit mass of consumers

interested in buying at most one of the base products. The base products supplied by D and T

are perfect substitutes. A consumer’s value for one of these products is denoted by v ∈ [0, 1].

We assume that v is distributed according to C.D.F. F (v) = v. Each consumer can be either

sophisticated or naïve and the probability of being naïve, which is independent of the willingness

to pay, is denoted by σ ∈ [0, 1). A sophisticated consumer understands that the add-on is worthless

whereas a naïve consumer can be “talked”into paying up to f for the add-on if he purchases the

base-good as well. The fringe supplies an imperfect substitute base good that consumers value at

vF = kv, with k ∈(0, 1− f

). The parameter k measures the substitutability of the base product

supplied by the fringe. Firms compete by simultaneously choosing prices for their base goods.

If retailer D is not committed to serve only naïve consumers, the unique equilibrium of the

pricing game takes the following form:

pF = pD = pT = 0 and fD = f.

19

With these prices, all consumers buy from retailer D and profits equal:

πF = πT = 0 and πD = σf.

Next, we look for an equilibrium where retailer D serves only naïve consumers. Firms in the

competitive fringe must make zero profits, hence p∗F = 0. Therefore, when buying from a firm in

the fringe, a consumer with type v obtains surplus equal to kv.

Under the presumed market segmentation, a sophisticated consumer will not be served by D.

Hence, a sophisticated consumer with valuation v will buy from T (rather than F ) if

v − pT ≥ kv ⇔ v ≥ pT1− k .

From the inequality above we immediately obtain the demand function of retailer T . Hence,

retailer T solves the following problem:

maxpT

[1− pT

1− k

]pT (1− σ) .

Taking the first-order condition and re-arranging yields

p∗T =1− k

2.

Retailer D, on the other hand, targets only naïve consumers. A naïve consumer with valuation

v will buy from D rather than F if

v − pD ≥ kv ⇔ v ≥ pD1− k .

Hence, firm D solves the following problem:

maxpD

[1− pD

1− k

] (pD + f

)σ.

Taking the first-order condition and re-arranging yields

p∗D =

(1− f

)− k

2.

Hence, we have the following result:

Proposition 5. Suppose retailer D is committed not to serve sophisticated consumers. Then, if

σ (1− k)2 ≤ f 2 there exists a Nash equilibrium of the pricing game with higher than Bertrand

20

profits for retailers D and T . The equilibrium prices and profits are

p∗F = 0, p∗D =1− k − f

2, p∗T =

1− k2

, f ∗D = f

and

π∗F = 0, π∗D =σ(1− k + f

)2

4 (1− k), π∗T =

(1− σ) (1− k)

4.

At these prices, all naïve consumers with v ∈[

1−k−f2(1−k)

, 1]buy fromD, all sophisticated consumers

with v ∈[

12, 1]buy from T and all remaining consumers buy from the fringe.16 Hence, market

segmentation still arises in equilibrium. Yet, the equilibrium prices and profits of D and T are

decreasing in k since, as the product supplied by the fringe becomes a better substitute, the

market power of D and T is reduced. Intuitively, the addition of a competitive fringe represents an

attractive outside option for some consumers; this, in turn, forces D and T to charge lower prices

and it reduces their profits compared to the model of Section 2. Notice, however, that the condition

for an equilibrium with market segmentation to exist is less restrictive than in the duopoly model

of Section 2 as the critical threshold on the fraction of naïve consumers is higher(

f2

(1−k)2> f 2

).

While this might appear counterintuitive at first, the intuition is that the incentives to deviate for

the transparent retailer are now weaker. Indeed, exactly because of the outside option represented

by the fringe, the gains for retailer T to undercut retailer D are reduced: the necessary price cut is

now relatively high compared to the lower markup. Nevertheless, as the product supplied by the

fringe is an imperfect substitute for the one supplied by D and T , these two retailers still retain

some market power which enables them to avoid the Bertrand trap. Hence, retailers D and T need

to retain suffi cient market power for our main result to extend beyond duopoly.17

4 Conclusion

The recent literature in Behavioral Industrial Organization has highlighted how consumer

naïveté affects firms’pricing and advertising strategies and how these, in turn, affect consumer

and total welfare in many different markets. Our paper contributes to this literature by showing

how firms’attempts to exploit consumer naïveté may have important implications for the design

of employees’compensation schemes. In particular, our analysis suggests that incentive schemes

that at first glance may appear counterproductive – like enforcing a target on add-on sales that

16It is easy to see that the chain company can design a compensation scheme, along the lines of the one derivedin Section 2, that would induce the manager of retailer D not to serve sophisticated consumers. For the sake ofbrevity we omit the details.17This is a feature shared by virtually all models with add-on pricing and shrouded attributes: with perfect (or

Bertrand) competition all add-on profits are competed away by reducing the base good’s price. In order to havean equilibrium with strictly positive profits, some authors have assumed product differentiation; see Ellison (2005),Dahremöller (2013) and Heidhues and Koszegi (2017). Others, instead, have introduced an exogenous price floorfor the base good; see Heidhues, Koszegi and Murooka (2016, 2017).

21

pushes employees to forgo a sale altogether rather than selling a product without an add-on –

may actually increase firms’profits. Moreover, our model delivers several new, interesting welfare

implications. First, naïve consumers might exercise a negative externality on rational consumers

who end up facing higher prices because of the formers. Second, welfare might be not monotone

in the fraction of naïve consumers so that educating/de-biasing naïve consumers could actually

lower total welfare in the market.

While we have framed our analysis in the context of a retail market for products like consumer

electronics, we believe that our model applies also to retail financial services such as credit cards,

insurance policies, and mortgages. Indeed, it is not uncommon for firms operating in this industry

to bundle basic financial products, like a checking account, together with expensive add-ons, like

an overdraft service, and offer them as an indivisible package.18 For instance, in what has become

known as the UK payment protection insurance mis-selling scandal, financial institutions sold

consumers credit lines together with payment protection insurance (PPI), also known as credit

insurance. In order to obtain the credit, consumers were often forced to purchase also the PPI.

The PPI was not only heftily priced but also often useless to the consumers as shown by the fact

that the fraction of rejected claims was very high compared to other types of insurance. Yet, bank

clerks had strong incentives to sell these products via huge commissions (Ferran, 2012).

An important assumption in our model is that the contract of the deceptive retailer’s manager

is observable to the other firms in the market as this makes the contract work as a credible

commitment device. We think this is a reasonable assumption since employment contracts usually

last for several years and cannot be adjusted as easily or frequently as prices. Nevertheless,

analyzing the case of unobservable (or imperfectly observable) contracts is an interesting question

left for future research.

Another interesting policy question is whether firms have an incentive to educate consumers

by disclosing/unshrouding information about the add-on. We have chosen not to directly model

this issue in our paper as the environment that we consider is a highly asymmetric one, with only

one deceptive firm in the market that can offer the add-on. One might conjecture that transparent

firms would have a strong incentive to educate consumers and warn them about the deceptive

firm’s add-on. Yet, as our analysis has shown, a transparent firm also benefits from the deceptive

firm’s bait-and-ditch strategy and the resulting market segmentation. Hence, our model suggests

that even transparent firms might prefer to keep naïve consumers in the dark.

18In the wake of the Wells Fargo fake accounts scandal in the US, initial reports blamed individual Wells Fargobranch managers for the problem, claiming that they gave their employees strong incentives to sell multiple financialproducts. This blame was later shifted to a pressure from higher-level management to open as many accounts aspossible through cross-selling – the practice of selling an additional product or service to an existing customer.

22

A ProofsProof of Proposition 1: The prices pD and pT constitute a Nash equilibrium only if no retailerhas an incentive to deviate. Under the presumption that the manager of retailer D is committednot to serve sophisticated consumers, there is no profitable deviation for him. Retailer T , on theother hand, is not committed to serve only sophisticated consumers. It could slightly undercut Dby offering the base good at price pT = pD − ε and serve both types of consumers. For ε → 0,retailer T’s profit from this deviation is

πDEVT =[1− 1

2

(1− f

)] 1

2

(1− f

)=

1

4

(1− f 2

)The deviation is not profitable if

1

4(1− σ) ≥ 1

4

(1− f 2

)⇐⇒ f 2 ≥ σ.

Hence, prices pD and pT constitute a Nash equilibrium of the pricing game. �Proof of Proposition 2: Suppose σ ≤ f 2. To prove that we have a subgame-perfect equilibriumwe need to show that no player has an incentive to deviate at each stage of the game. We knowfrom Proposition 1 that retailer T has no incentive to deviate if σ ≤ f 2. Next, we need to show thatfixing retailer T’s strategy and the contract signed between the chain company and the managerof D, the latter does not want to deviate. Recall that the compensation scheme offered by thechain company is:

w(rB, rA) = min {rA, rA}+ min {rB, rB} − F,

where rB = σ4(1− f 2), rA = σ

2(1 + f)f and F = rB + rA. Given this scheme, if the manager follows

the presumed equilibrium strategy of charging pD = pD, fD = fD, and not serving sophisticatedconsumers, his utility is exactly zero. The manager of D could deviate by serving sophisticatesat the presumed equilibrium prices and/or by changing the prices as well. Yet, any deviation is(weakly) dominated. Indeed, if he were to raise a revenue from add-on (resp. base good) saleslower than rA (resp. rB), the manager would attain a strictly negative payoff. On the other hand,if he were to raise a higher revenue on either product, his compensation would not increase. Hence,there are no profitable deviations for the manager of store D.Finally, we need to show that at the first stage the manager is willing to accept the proposed

contract and that the chain company cannot do better by offering a different contract. First, giventhat the manager’s outside option is U = 0, he is indifferent between rejecting the contract andaccepting it. Next, notice that any contract that induces the manager to maximize downstreamprofits would result in Bertrand pricing for the base good so that the chain’s profits would beΠ = σf . By offering the contract w(rB, rA), instead, the chain’s profits are Π = σ

4(1 + f)2. We

have that σ

4(1 + f)2 > σf ⇔

(1− f

)2> 0.

Therefore, we have that the proposed strategy profile constitutes a subgame-perfect equilibrium.A final remark regarding equilibrium existence is in order. With the contract space being

unrestricted, existence of a subgame-perfect equilibrium cannot be guaranteed. For instance, ifthe agent’s wage is increasing in rB for rB strictly smaller than a certain threshold, then the agent’s

23

best response is not well defined. In order to avoid this issue, we have to assume that the wagepayment can be contingent only on a discrete set of revenues R = {(rA, rB)} that always includes(rA, rB). For any |R| > 1 —in particular for |R| → ∞ —equilibrium existence is guaranteed andthe outcome described in the proposition is an equilibrium outcome. �Proof of Proposition 3: First, it is easy to see that when retailer D is a monopolist, it isoptimal to charge fmD = f for the add on. Then, for the base good, retailer D chooses the pricethat maximizes the following expression

πmD(pD) = (1− pD)(pD + σf

).

Taking the first-order condition and re-arranging yields

pmD =1− σf

2.

Hence, profits and consumer welfare in the market equal

πmD =

(1 + σf

)2

4and CSm = σ

∫ 1

pmD

(v − pmD − f

)dv + (1− σ)

∫ 1

pmD

(v − pmD) dv =1− 2fσ − 3f 2σ2

8.

On the other hand, under duopoly and bait-and-ditch we have

πT =1− σ

4, πD =

σ

4(1+f)2 and CS = σ

∫ 1

pD

(v − pD − f

)dv+(1− σ)

∫ 1

pT(v − pT ) dv =

1− 2fσ − 3f 2σ

8.

It is easy to verify that CS < CSm and πmD < πT + πD. The result then follows since the overallmeasure of consumers who buy is the same under both scenarios; yet, more sophisticates buy whenwhen retailer D is a monopolist. In other words, when moving from monopoly to duopoly withbait-and-ditch, consumption is shifted away from high-value sophisticated consumers to low valuenaïve consumers. �In the proof of Proposition 4 we use the two following results:

Lemma 1. Suppose the chain wants to serve both types of consumers. Then, it induces its agentto exert high effort if and only if ψ ≤ ψ, with

ψ :=(H − L)(q1 − q0)2f

q1

.

Proof of Lemma 1: The optimal contracts for e = 0 and e = 1 are derived in the main text aswell as the corresponding profits. The chain prefers to induce high effort if and only if Π1 ≥ Π0,which is equivalent to

ψ ≤ (H − L)(q1 − q0)2f

q1

=: ψ.

Hence, the stated result follows. �

24

Lemma 2. Suppose the chain wants to serve only naïve consumers. Then, it induces its agent toexert high effort if and only if ψ ≤ ψ, with

ψ :=(H − L)(q1 − q0)2

q1

(1 + f)2

4+

(q1 − q0

q1

)φ.

Proof of Lemma 2: First, suppose the chain wants to induce low effort, e = 0. As explained inthe main text, in this case the optimal wages are:

wH,S = wL,S = 0 and wH,N = wL,N = φ.

The chain’s corresponding profit is

Π0 =σ0(1 + f)2

4− φ.

Now, suppose the chain wants to induce e = 1. It is easy to verify that under the optimalcontract we have

wH,S = wL,S = 0.

Moreover, if (PCN1 ) holds, (ICN1,N) and (IC

NN) are automatically satisfied. The remaining con-

straints are:

wL,N + q1(wH,N − wL,N) ≥ φ+ ψ (PCN1 )

(q1 − q0)(wH,N − wL,N) ≥ ψ (ICN1 )

wH,N ≥ 0 wL,N ≥ 0 (LL)

The chain wants to minimize the expected wage payment. Thus, the constraint (ICN1 ) willalways be binding. The question is whether (LL) or (PCN1 ) is slack. If (LL) does not bind, weobtain

wL,N = φ− ψ q0

q1 − q0

wH,N = φ+ ψ1− q0

q1 − q0

Under Assumption 3 we have that wL,N < 0 and thus constraint (LL) is violated. Hence, underthe optimal contract, the limited liability constraint is binding while the participation constraintis slack. Formally,

wL,N = wH,S = wL,S = 0, wH,N =ψ

q1 − q0

.

The chain’s profit in this case is

Π1 =σ1(1 + f)2

4− q1

q1 − q0

ψ.

25

The chain prefers to induce high effort; i.e., Π1 ≥ Π0, if and only if

ψ ≤ (H − L)(q1 − q0)2

q1

(1 + f)2

4+q1 − q0

q1

φ =: ψ.

This concludes the proof of the lemma. �Proof of Proposition 4: From the above two lemmas it follows that we can distinguish threecases depending on the size of ψ —cases (i) - (iii) from the proposition.Part (i) of the proposition: The claim is true if and only if Π1 < Π1, which is equivalent to

σ1f −q1

q1 − q0

ψ >σ1(1 + f)2

4− q1

q1 − q0

ψ

⇐⇒ 0 <σ1(1− f)2

4.

Next, we prove part (ii) of the proposition. Note that Π0 ≤ Π1 is equivalent to

σ0f ≤σ1(1 + f)2

4− q1

q1 − q0

ψ

⇐⇒ q1

q1 − q0

ψ ≤ σ0(1− f)2

4+ (q1 − q0)(H − L)

(1 + f)2

4.

By multiplying the both sides of the above inequality by (q1− q0)/q1, we obtain the inequalitydisplayed in the proposition.Finally, in case (iii), not serving sophisticated consumers is optimal if and only if Π0 < Π0.

This is equivalent to

σ0f <σ0(1 + f)2

4− φ

⇐⇒ φ <σ0(1− f)2

4.

which completes the proof. �Proof of Proposition 5: The prices p∗F , p

∗D and p

∗T constitute a Nash equilibrium of the pricing

game only if no retailer has an incentive to deviate. Firms in the fringe cannot deviate because theyhave to make zero profits. Under the presumption that the manager of retailer D is committednot to serve sophisticated consumers, there is no profitable deviation for him either. Retailer T , onthe other hand, is not committed to serve only sophisticated consumers. It could slightly undercutD by offering the base good at price pT = p∗D − ε and serve both types of consumers. For ε→ 0,retailer T’s profit from this deviation is

πDEVT =

[1− 1− k − f

2 (1− k)

]1− k − f

2

=(1− k)2 − f 2

4 (1− k).

26

The deviation is not profitable if

1− σ1− k

(1− k

2

)2

≥ (1− k)2 − f 2

4 (1− k)⇐⇒ f 2 ≥ σ (1− k)2 .

Hence, prices p∗F , p∗D and p

∗T constitute a Nash equilibrium of the pricing game. �

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